See for instance Eq. (2.13) in the review article [1]}.
In this notation,
recalling that \(\widehat{m}_0=\widehat{X}_0\)  we have the bias formulae
\(\widehat{m}_t-\widehat{X}_t=\int _0^t {\cal E }_{s,t}~\underbrace{(\widehat{p}_s-P_s)}_{\in [-c_1/N,0]~a.e.}~C^{\prime }\Sigma ^{-1}~\left(dY_s-C\widehat{X}_s~dt\right)+\int _0^t {\cal E }_{s,t}~\mathbb {E}\left((P_s-p_s)~S(m_s-\widehat{X}_s)~|~ {\cal F }_s\right)~ds.\) 
